Hyperbola with Shifted Centre

Centre at (h, k). Identified by completing the square. All elements of the standard hyperbola apply with origin shifted to (h, k).

For a hyperbola with centre at $(h, k)$, substitute $X = x-h$, $Y = y-k$:

$$\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1$$

Identifying from a general equation: $Ax^2 - Cy^2 + Dx + Ey + F = 0$ (with $A, C > 0$, no $xy$ term) represents a hyperbola. Complete the square:

$$A(x-h)^2 - C(y-k)^2 = K$$

Divide by $K$. If $K > 0$: transverse axis parallel to x-axis; if $K < 0$: parallel to y-axis.

Elements in original coordinates for $(x-h)^2/a^2 - (y-k)^2/b^2 = 1$: